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Mirrors > Home > ILE Home > Th. List > eufnfv | Unicode version |
Description: A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.) |
Ref | Expression |
---|---|
eufnfv.1 | |
eufnfv.2 |
Ref | Expression |
---|---|
eufnfv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eufnfv.1 | . . . . 5 | |
2 | 1 | mptex 5614 | . . . 4 |
3 | eqeq2 2127 | . . . . . 6 | |
4 | 3 | bibi2d 231 | . . . . 5 |
5 | 4 | albidv 1780 | . . . 4 |
6 | 2, 5 | spcev 2754 | . . 3 |
7 | eufnfv.2 | . . . . . . 7 | |
8 | eqid 2117 | . . . . . . 7 | |
9 | 7, 8 | fnmpti 5221 | . . . . . 6 |
10 | fneq1 5181 | . . . . . 6 | |
11 | 9, 10 | mpbiri 167 | . . . . 5 |
12 | 11 | pm4.71ri 389 | . . . 4 |
13 | dffn5im 5435 | . . . . . . 7 | |
14 | 13 | eqeq1d 2126 | . . . . . 6 |
15 | funfvex 5406 | . . . . . . . . 9 | |
16 | 15 | funfni 5193 | . . . . . . . 8 |
17 | 16 | ralrimiva 2482 | . . . . . . 7 |
18 | mpteqb 5479 | . . . . . . 7 | |
19 | 17, 18 | syl 14 | . . . . . 6 |
20 | 14, 19 | bitrd 187 | . . . . 5 |
21 | 20 | pm5.32i 449 | . . . 4 |
22 | 12, 21 | bitr2i 184 | . . 3 |
23 | 6, 22 | mpg 1412 | . 2 |
24 | df-eu 1980 | . 2 | |
25 | 23, 24 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wal 1314 wceq 1316 wex 1453 wcel 1465 weu 1977 wral 2393 cvv 2660 cmpt 3959 wfn 5088 cfv 5093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 |
This theorem is referenced by: (None) |
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